What is the Limit of a Rational Function?

[efekwulsemmay]

Homework Statement

All the problem is asking me to do is find the limit. I have to do it algebraicly too which sucks. I can't figure out how to do it.
\lim_{x\rightarrow0} \frac{\frac{1}{x+2}-\frac{1}{2}}{x}

Homework Equations

I am not sure really.

The Attempt at a Solution

One thing I have tried is to multiply by \frac {x}{x}. Which gave me:
\lim_{x\rightarrow0} \frac{\frac{1}{x+2}-\frac{1}{2}}{x} \times \frac {x}{x}

=\frac{\frac{1\times x}{x+2}-\frac{1\times x}{2}}{x\times x}

= \frac{\frac{x}{x+2}-\frac{x}{2}}{x^{2}}

=\frac{x}{x+2}\rightarrow \frac{x}{x}+\frac{x}{2} \rightarrow 1+\frac{x}{2}

=\frac{1+\frac{x}{2}-\frac{x}{2}}{x^{2}}

=\frac{1}{x^{2}}

1) I am not sure of my algebra during this and
2) I don't know where to go from here should my algebra check out.

I have already tried to multiply by \frac{\sqrt{x}}{\sqrt{x}} but it just seems to give me \frac{\sqrt{x}}{x} which doesn't help. I am stuck help me please?

 

[Mark44]

Combine the two fractions in the numerator and then simplify the whole thing.

 

[efekwulsemmay]

Thank you! :):):):):):):):)